Optimal. Leaf size=402 \[ -\frac{14 i b^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{15 c^3}-\frac{1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{11 i a b d^3 x}{6 c^2}-\frac{37 i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{20 c^3}-\frac{28 b d^3 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{15 c^3}+\frac{3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{11}{18} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{14 b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac{113 i b^2 d^3 \log \left (c^2 x^2+1\right )}{90 c^3}+\frac{37 b^2 d^3 x}{30 c^2}+\frac{11 i b^2 d^3 x \tan ^{-1}(c x)}{6 c^2}-\frac{37 b^2 d^3 \tan ^{-1}(c x)}{30 c^3}-\frac{1}{60} i b^2 c d^3 x^4+\frac{61 i b^2 d^3 x^2}{180 c}-\frac{1}{10} b^2 d^3 x^3 \]
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Rubi [A] time = 1.19617, antiderivative size = 402, normalized size of antiderivative = 1., number of steps used = 52, number of rules used = 15, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4876, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 266, 43, 4846, 260, 4884, 302} \[ -\frac{14 i b^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{15 c^3}-\frac{1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{11 i a b d^3 x}{6 c^2}-\frac{37 i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{20 c^3}-\frac{28 b d^3 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{15 c^3}+\frac{3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{11}{18} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{14 b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac{113 i b^2 d^3 \log \left (c^2 x^2+1\right )}{90 c^3}+\frac{37 b^2 d^3 x}{30 c^2}+\frac{11 i b^2 d^3 x \tan ^{-1}(c x)}{6 c^2}-\frac{37 b^2 d^3 \tan ^{-1}(c x)}{30 c^3}-\frac{1}{60} i b^2 c d^3 x^4+\frac{61 i b^2 d^3 x^2}{180 c}-\frac{1}{10} b^2 d^3 x^3 \]
Antiderivative was successfully verified.
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Rule 4876
Rule 4852
Rule 4916
Rule 321
Rule 203
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 266
Rule 43
Rule 4846
Rule 260
Rule 4884
Rule 302
Rubi steps
\begin{align*} \int x^2 (d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\int \left (d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+3 i c d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-3 c^2 d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-i c^3 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^3 \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+\left (3 i c d^3\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx-\left (3 c^2 d^3\right ) \int x^4 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx-\left (i c^3 d^3\right ) \int x^5 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{3} \left (2 b c d^3\right ) \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac{1}{2} \left (3 i b c^2 d^3\right ) \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac{1}{5} \left (6 b c^3 d^3\right ) \int \frac{x^5 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac{1}{3} \left (i b c^4 d^3\right ) \int \frac{x^6 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{2} \left (3 i b d^3\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx+\frac{1}{2} \left (3 i b d^3\right ) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac{\left (2 b d^3\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c}+\frac{\left (2 b d^3\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c}+\frac{1}{5} \left (6 b c d^3\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx-\frac{1}{5} \left (6 b c d^3\right ) \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac{1}{3} \left (i b c^2 d^3\right ) \int x^4 \left (a+b \tan ^{-1}(c x)\right ) \, dx-\frac{1}{3} \left (i b c^2 d^3\right ) \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=-\frac{b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac{1}{2} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{3} \left (i b d^3\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx+\frac{1}{3} \left (i b d^3\right ) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac{1}{3} \left (b^2 d^3\right ) \int \frac{x^2}{1+c^2 x^2} \, dx+\frac{\left (3 i b d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c^2}-\frac{\left (3 i b d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 c^2}-\frac{\left (2 b d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{3 c^2}-\frac{\left (6 b d^3\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{5 c}+\frac{\left (6 b d^3\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{5 c}+\frac{1}{2} \left (i b^2 c d^3\right ) \int \frac{x^3}{1+c^2 x^2} \, dx-\frac{1}{10} \left (3 b^2 c^2 d^3\right ) \int \frac{x^4}{1+c^2 x^2} \, dx-\frac{1}{15} \left (i b^2 c^3 d^3\right ) \int \frac{x^5}{1+c^2 x^2} \, dx\\ &=\frac{3 i a b d^3 x}{2 c^2}+\frac{b^2 d^3 x}{3 c^2}-\frac{14 b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac{11}{18} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{101 i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{60 c^3}+\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{2 b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^3}+\frac{1}{5} \left (3 b^2 d^3\right ) \int \frac{x^2}{1+c^2 x^2} \, dx+\frac{\left (i b d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c^2}-\frac{\left (i b d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{3 c^2}-\frac{\left (6 b d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{5 c^2}+\frac{\left (3 i b^2 d^3\right ) \int \tan ^{-1}(c x) \, dx}{2 c^2}-\frac{\left (b^2 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx}{3 c^2}+\frac{\left (2 b^2 d^3\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^2}+\frac{1}{9} \left (i b^2 c d^3\right ) \int \frac{x^3}{1+c^2 x^2} \, dx+\frac{1}{4} \left (i b^2 c d^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )-\frac{1}{10} \left (3 b^2 c^2 d^3\right ) \int \left (-\frac{1}{c^4}+\frac{x^2}{c^2}+\frac{1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx-\frac{1}{30} \left (i b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+c^2 x} \, dx,x,x^2\right )\\ &=\frac{11 i a b d^3 x}{6 c^2}+\frac{37 b^2 d^3 x}{30 c^2}-\frac{1}{10} b^2 d^3 x^3-\frac{b^2 d^3 \tan ^{-1}(c x)}{3 c^3}+\frac{3 i b^2 d^3 x \tan ^{-1}(c x)}{2 c^2}-\frac{14 b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac{11}{18} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{37 i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{20 c^3}+\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{28 b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{15 c^3}-\frac{\left (2 i b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{3 c^3}+\frac{\left (i b^2 d^3\right ) \int \tan ^{-1}(c x) \, dx}{3 c^2}-\frac{\left (3 b^2 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx}{10 c^2}-\frac{\left (3 b^2 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx}{5 c^2}+\frac{\left (6 b^2 d^3\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{5 c^2}-\frac{\left (3 i b^2 d^3\right ) \int \frac{x}{1+c^2 x^2} \, dx}{2 c}+\frac{1}{18} \left (i b^2 c d^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )+\frac{1}{4} \left (i b^2 c d^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{1}{30} \left (i b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}+\frac{x}{c^2}+\frac{1}{c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{11 i a b d^3 x}{6 c^2}+\frac{37 b^2 d^3 x}{30 c^2}+\frac{17 i b^2 d^3 x^2}{60 c}-\frac{1}{10} b^2 d^3 x^3-\frac{1}{60} i b^2 c d^3 x^4-\frac{37 b^2 d^3 \tan ^{-1}(c x)}{30 c^3}+\frac{11 i b^2 d^3 x \tan ^{-1}(c x)}{6 c^2}-\frac{14 b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac{11}{18} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{37 i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{20 c^3}+\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{28 b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{15 c^3}-\frac{31 i b^2 d^3 \log \left (1+c^2 x^2\right )}{30 c^3}-\frac{i b^2 d^3 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{3 c^3}-\frac{\left (6 i b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{5 c^3}-\frac{\left (i b^2 d^3\right ) \int \frac{x}{1+c^2 x^2} \, dx}{3 c}+\frac{1}{18} \left (i b^2 c d^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{11 i a b d^3 x}{6 c^2}+\frac{37 b^2 d^3 x}{30 c^2}+\frac{61 i b^2 d^3 x^2}{180 c}-\frac{1}{10} b^2 d^3 x^3-\frac{1}{60} i b^2 c d^3 x^4-\frac{37 b^2 d^3 \tan ^{-1}(c x)}{30 c^3}+\frac{11 i b^2 d^3 x \tan ^{-1}(c x)}{6 c^2}-\frac{14 b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac{11}{18} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{37 i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{20 c^3}+\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{28 b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{15 c^3}-\frac{113 i b^2 d^3 \log \left (1+c^2 x^2\right )}{90 c^3}-\frac{14 i b^2 d^3 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{15 c^3}\\ \end{align*}
Mathematica [A] time = 1.3733, size = 369, normalized size = 0.92 \[ \frac{d^3 \left (168 i b^2 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )-30 i a^2 c^6 x^6-108 a^2 c^5 x^5+135 i a^2 c^4 x^4+60 a^2 c^3 x^3+12 i a b c^5 x^5+54 a b c^4 x^4-110 i a b c^3 x^3-168 a b c^2 x^2+168 a b \log \left (c^2 x^2+1\right )+2 b \tan ^{-1}(c x) \left (3 a \left (-10 i c^6 x^6-36 c^5 x^5+45 i c^4 x^4+20 c^3 x^3-55 i\right )+b \left (6 i c^5 x^5+27 c^4 x^4-55 i c^3 x^3-84 c^2 x^2+165 i c x-111\right )-168 b \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )+330 i a b c x-162 a b-3 i b^2 c^4 x^4-18 b^2 c^3 x^3+61 i b^2 c^2 x^2-226 i b^2 \log \left (c^2 x^2+1\right )+3 b^2 (c x-i)^4 \left (-10 i c^2 x^2+4 c x+i\right ) \tan ^{-1}(c x)^2+222 b^2 c x+64 i b^2\right )}{180 c^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.1, size = 712, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{240} \,{\left (10 i \, b^{2} c^{3} d^{3} x^{6} + 36 \, b^{2} c^{2} d^{3} x^{5} - 45 i \, b^{2} c d^{3} x^{4} - 20 \, b^{2} d^{3} x^{3}\right )} \log \left (-\frac{c x + i}{c x - i}\right )^{2} +{\rm integral}\left (\frac{-60 i \, a^{2} c^{5} d^{3} x^{7} - 180 \, a^{2} c^{4} d^{3} x^{6} + 120 i \, a^{2} c^{3} d^{3} x^{5} - 120 \, a^{2} c^{2} d^{3} x^{4} + 180 i \, a^{2} c d^{3} x^{3} + 60 \, a^{2} d^{3} x^{2} +{\left (60 \, a b c^{5} d^{3} x^{7} +{\left (-180 i \, a b - 10 \, b^{2}\right )} c^{4} d^{3} x^{6} - 12 \,{\left (10 \, a b - 3 i \, b^{2}\right )} c^{3} d^{3} x^{5} +{\left (-120 i \, a b + 45 \, b^{2}\right )} c^{2} d^{3} x^{4} - 20 \,{\left (9 \, a b + i \, b^{2}\right )} c d^{3} x^{3} + 60 i \, a b d^{3} x^{2}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{60 \,{\left (c^{2} x^{2} + 1\right )}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, c d x + d\right )}^{3}{\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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